This activity involves rounding four-digit numbers to the nearest thousand.

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these numbers to the nearest whole number?

Delight your friends with this cunning trick! Can you explain how it works?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Here are two kinds of spirals for you to explore. What do you notice?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

This task follows on from Build it Up and takes the ideas into three dimensions!

An investigation that gives you the opportunity to make and justify predictions.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Can you find all the ways to get 15 at the top of this triangle of numbers?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.